Ukip leadership favourite Steven Woolfe has called on his party to stop engaging in “politics of the sixth form college” if it wants to get serious about winning elections.

Woolfe, a former City lawyer, said he would take steps to end the infighting that has dogged the party over the last year and declared he would review all party policies set at the last election under Nigel Farage.

The party’s immigration spokesman is the frontrunner in a crowded field of contenders to be Ukip leader, including Lisa Duffy, a key party organiser, Jonathan Arnott, an MEP in the north-east, and Bill Etheridge, an MEP in the Midlands.

In an interview with the Guardian, Woolfe said he was determined to champion social mobility and appeal to voters on the left, denying that Ukip was a rightwing political party. But he said the party had work to do to transform itself into a sharper political force that would pose a serious challenge to Labour in the north and the Tories in the south.

Since the election last year, Ukip has been gripped by internal feuding that put Farage at odds with the party’s only MP, Douglas Carswell, and its former deputy chair, Suzanne Evans.

Woolfe said: “I will recognise that on occasions, Ukip has in some ways seemed to reflect the politics of the sixth form college and we are in no way ever going to succeed seriously at the ballot box and win seats as parliamentarians in large numbers of council seats if we maintain that attitude.”

Woolfe said he did not blame Farage for letting the party descend into civil war, but those around him for failing to manage the party effectively. “Nigel’s great skill is that of an orator and person who can carry a message out as he has done successfully ... He left it to those in managerial positions. There is a lot to be said for those people not being capable in actually maintaining the party,” he said.

There has been speculation that Woolfe could join the ranks of senior Ukip politicians such as Arron Banks and former MP Mark Reckless in being excluded from running for the leadership because they had not been a member for the required five years. Leaked emails suggest he failed to pay his membership for 15 months, meaning it lapsed, but he dismissed talk of his bid being derailed by the issue as “absolute nonsense”.

Woolfe said he would be happy to get an endorsement from Farage but also highlighted differences, including his own upbringing on a council estate in Mosside, Manchester, and his mixed-race heritage.

The MEP said he would prioritise holding the government’s feet to the fire to ensure a full exit from the EU, controlling immigration and fighting for more grammar schools in 50 of the UK’s poorest areas, which he claimed fostered social mobility.

He cited Ukip’s opposition to the bedroom tax as evidence that it was not a rightwing party but suggested he would like a lower-tax society, cuts to corporation tax and less regulation for small businesses.

Asked what the top rate of tax should be, he said Ukip would put the top rate at 45% but added: “We will have a review of that again. We will have a look at our manifesto. I know it seems only a year or so ago, but the government has made changes and I think we need to do so too.”

Duffy, the second favourite to win the contest who is backed by Evans, launched her leadership bid in London on Thursday. She told the Guardian’s politics weekly podcast that she would solve “troubles at the top of the party”.

“The difference between myself and Nigel is that I see the future of Ukip as a team-based future. I want to make sure we use all the talent in our party and use all the grassroots because they are the backbone,” Duffy said.

She said there was a “very big job in terms of making sure we get the right Brexit deal” but that the party also had to start talking about contentious issues such as “integration, showing a positive vision of British Islam, about keeping the pressure on and making sure our NHS is well funded and taking a tough line on law and order”.

Asked what she meant by the positive approach to Islam, she said: “There are a lot of British Muslims in our country and obviously there is a lot of extremism. We need to work with the Muslim communities and eradicate the extremism because most Muslims are good, hardworking, honest people, and it is extremism giving them a bad name. We need to work and make sure there is a positive vision. It needs to be put together with experts from within the Muslim community and in our party.”